Download ON ∗-AUTONOMOUS CATEGORIES OF TOPOLOGICAL

ON ∗-AUTONOMOUS CATEGORIES OF TOPOLOGICAL MODULES
MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL
Abstract. Let R be a commutative ring whose complete ring of quotients is Rinjective. We show that the category of topological R-modules contains a full subcategory that is ∗-autonomous using R itself as dualizing object. In order to do this, we
develop a new variation on the category chu(D , R), where D is the category of discrete
R-modules: the high wide subcategory, which we show equivalent to the category of
reflexive topological modules.
1. Introduction
Let R be a ring (with unit). In [Barr, et. al. (2009)], we showed that under certain reasonable conditions on R, a full subcategory of the category of right R-modules is equivalent
to a full subcategory of the category of topological left R-modules. These conditions are
explained in detail in the cited paper. When R is commutative, the conditions simplify
and reduce to the assumption that the complete ring of quotients Q of R, as described in
[Lambek (1986), Section 3] be R-injective (op. cit., Proposition 4.3.3). It is sufficient, but
far from necessary, that R have no non-zero nilpotents. When K is a field and p ∈ K[x]
is any polynomial, the residue ring R = K[x]/(p) will always be its own complete ring of
quotients and also be self-injective. But if p has repeated factors, R will have nilpotents.
The main purpose of this paper is to show that when the commutative ring R satisfies
the injectivity condition of the preceding paragraph, then the category of topological Rmodules contains a full subcategory with both an autoduality and an internal hom. Such
a category is called ∗-autonomous, see, for example, [Barr (1999)]. As usually happens,
exhibiting such a structure requires a detour through the Chu construction (op. cit.).
However, since we are not supposing that R is self-injective, neither the full Chu category
nor the separated extension subcategory does the job and we are forced to introduce yet
another variation of the Chu category, that we call the high, wide subcategory (Definition
4.8).
We also consider the case that R is self-injective. Then all Chu objects are high
and wide and we do not need to introduce that subcategory; moreover, there will be
two distinct ∗-autonomous subcategories of topological R-modules. Although the two
categories are equivalent as categories—even as ∗-autonomous categories—one of them
contains all the discrete modules and the other one doesn’t.
The first and third authors would like to thank NSERC of Canada for its support of this research.
We would all like to thank McGill and Concordia Universities for partial support of the second author’s
visits to Montreal.
c Michael Barr, John F. Kennison, and R. Raphael, 2009. Permission to copy for private use granted.
°
1
2
2. The ring R
All objects we study in this note are modules over a commutative ring R of which we
make one further assumption: that the complete ring of quotients of R be R-injective (for
which it is necessary and sufficient that the complete ring of quotients be self-injective).
An ideal I of a commutative ring R is called dense if whenever 0 6= r ∈ R, then
rI 6= 0. The complete ring of quotients Q of R is characterized by the fact that it is
an essential extension of R and every homomorphism from a dense ideal to Q can be
/ Q. Details are found in [Lambek (1986), Sections 2.3
extended to a homomorphism R
and 4.3].
It is worth going into a bit more detail about the reference [Lambek (1986)]. In section
2.3, the complete ring of quotients for commutative rings is constructed, while in 4.3 the
construction is carried out in the non-commutative case, where the definition of “dense”
is more complicated. Unfortunately, all the discussion of injectivity is carried out in
the latter section and it is not easy to work out reasonable conditions under which the
complete ring of quotients is injective. Here is one simple case, although far from the only
one.
2.1. Example. Let K be a field. Each of the rings K[x]/(x)2 and K[x, y]/(x, y)2 , can
be readily seen to be its own complete rings of quotients (they have no proper dense
ideals), but the first is and the second is not self-injective. In the second case the ideal
(x, y) contains every proper ideal (and is therefore large, as defined in the third paragraph
below), but is not dense.
Let A be an R-module. An element a ∈ A is called a weak torsion element if there
is a dense ideal I ⊆ R with aI = 0. We say that A is weak torsion module if every
element of A is and that A is weak torsion free if it contains no non-zero weak torsion
elements.
An R-module is said to be R-cogenerated if it can be embedded into a power of R.
A topological R-module is called R-cogenerated if it can be embedded algebraically and
topologically into a power of R, with R topologized discretely. Among other things, this
implies that the topology is generated by (translates of) the open submodules.
An ideal I ⊆ R is called large if its intersection with every non-zero ideal is non-zero.
An obvious Zorn’s lemma argument shows that if every map from a large ideal of R to
/ Q, then this is true for all ideals and so
a module Q can be extended to a map R
Q is R-injective. A dense ideal is characterized by the fact that its product with every
non-zero ideal is non-zero. If, for example, there are no nilpotents, then when the product
of two ideals is non-zero, so is their intersection and then the complete ring of quotients
is injective. But while this condition is sufficient, it is far from necessary as the example
K[x]/(x)2 makes clear.
3
3. The category C
Let C denote the category of R-cogenerated topological R-modules. When C is an object
of C , we let |C| denote the discrete module underlying C and let ||C|| denote the discrete
set underlying C. If C and C 0 are objects of C , we let hom(C, C 0 ) denote the R-module
of continuous R-linear homomorphisms and Hom(C, C 0 ) denote the set ||hom(C, C 0 )||.
We denote by C ? the module hom(C, R) topologized as a subspace of R||C|| .
3.1.
Proposition. For C ∈ C , the canonical map C
/R||C ? || is a topological embedding.
Ä
Proof. Let C Â / RS define the topology on C. Then for each s ∈ S, the composite
/ RS ps / R, where ps is the projection, defines a homomorphism C
/ R. Thus
C
/ Hom(R, C) that leads to the commutative triangle
there is a function S
Ä
C DÂ
DD
DD
DD
DD
DD
DD
D"
/ RS
O
RHom(C,R)
and the diagonal map, being an initial factor of a topological embedding, is one itself.
For later use, we prove a theorem of a purely topological nature. It is true for arbitrary
topological R-modules, not just for objects of C .
Q
3.2. Theorem. Suppose A ⊆ s∈S As is an embedding of topological modules and D is
/ D factors through a quotient of A
a discrete module. Then
¡Q every¢ continuous map A
of the form A/(A ∩
s∈S−S0 )As where S0 is a finite subset of S.
/ D is given, then ker f must be open. A subbasic open neighbourhood
Proof. If f : A
³
´
Q
of 0 in the product consists of a set of the form A ∩ U × s6=s1 As with U open in
As1 .³ In particular,
every subbasic open neighbourhood at 0 includes a set of the form
´
Q
A∩
s6=s1 As and every basic neighbourhood of 0, being a finite intersection of subbasic
¡Q
¢
sets, contains a set of the form A ∩
As which implies that for some choice of a
s
∈S
/
0
¡Q
¢
finite set S0 , f vanishes on A ∩
s∈S0 As .
In fact, by analyzing this argument we find that hom(−, D) commutes with filtered
colimits (a product is a filtered colimit of the finite products) whenever D contains a
neighbourhood of 0 that contains no non-zero submodule. However, we require only the
form stated here.
3.3. Corollary. For any family
modules
and any discrete object D,
Q
P {As } of topoogical
/
hom( As , D) is an isomorphism.
we have that the canonical map
hom(As , R)
4
Crucial to this paper is the following theorem, which is proved in [Barr, et. al. (2009),
Corollary 3.8]. It is understood that R always carries the discrete topology. Incidentally,
this is the only place that the injectivity condition is used.
Ä
3.4. Theorem. Let C Â / C 0 be an algebraic and topological inclusion between objects
/ hom(C, R) is weak torsion.
of C . Then the cokernel of hom(C 0 , R)
4. The chu category
We begin this section with a brief description of the Chu categories and chu categories
that concern us in this article. They are instances of a much more general construction.
For more details, see [Barr (1998), Barr (1999)].
4.1. Definition. We denote by D the category of (discrete) R-modules. We define
a category Chu(D , R) as follows. An object of Chu(D , R) is a pair (A, X) in which
/ R. A morphism
A and X are R-modules together with a pairing h−, −i : A ⊗ X
/
/
/X
(f, g) : (A, X)
(B, Y ) consists of R-linear homomorphisms f : A
B and g : Y
such that hf a, yi = ha, gyi for a ∈ A and y ∈ Y . The definition of morphism is equivalent
to the commutativity of either of the squares
A
f
/ hom(X, R)
Y
²
hom(g,R)
B
²
/ hom(Y, R)
g
/ hom(B, R)
hom(f,R)
²
X
(∗)
²
/ hom(A, R)
in which the horizontal arrows are the adjoint transposes of the two h−, −i.
One way of getting an object of Chu(D , R) is to begin with a topological R-module C
and forming the pair (|C|, hom(C, R)) with the pairing given by evaluation. Then, if C and
C 0 are topological modules, it follows immediately that every continuous homomorphism
/ (|C 0 |, hom(C 0 , R)).
from C to C 0 induces a morphism (|C|, hom(C, R))
If U = (A, X) is an object of Chu(D , R), we denote by U⊥ the object (X, A) with
the evident pairing. If U = (A, X) and V = (B, Y ) are objects of Chu(D , R), the set of
/ V is obviously an R-module that we denote [U, V]. Then Chu(D , R)
morphisms U
becomes a ∗-autonomous category when we define
U −◦ V = ([U, V], A ⊗ Y )
with pairing h(f, g), (a, y)i = hf a, yi = ha, gyi. When R = (R, R) with the R-module
structure as pairing, one easily sees that U −◦ R = U⊥ . We call R the dualizing object.
There is also a tensor product given by
U ⊗ V = (U −◦ V⊥ )⊥
5
/ hom(X, R) is
We say that the object (A, X) is separated if the induced map A
/
monic and that it is extensional if X
hom(A, R) is monic. (Incidentally, extensionality
is the property of functions that two are equal if they are equal for all possible arguments.
/ R).
Thus extensionality here means that, in effect, X is a module of homomorphisms A
A pair is called non-singular if it is both separated and extensional. This means that
for all 0 6= a ∈ A there is an x ∈ X with ha, xi 6= 0 and, symmetrically, that for all
0 6= x ∈ X, there is an a ∈ A with ha, xi 6= 0. The results in the theorem that follows are
proved in detail in [Barr (1998)]. Since D is abelian the factorization referred to in that
citation can only be the standard one into epics and monics. Let Chus (D , R), Chue (D , R),
and chu(D , R) denote, respectively, the full subcategories of Chu(D , R) consisting of the
separated, the extensional, and the separated extensional objects.
4.2.
Theorem. [Barr (1998)]
1. The inclusion Chus (D , R) Â
2. the inclusion Chue (D , R) Â
Ä
Ä
/ Chu(D , R) has a left adjoint S;
/ Chu(D , R) has a right adjoint E;
3. SE ∼
= ES;
4. when (A, X) is extensional and (B, Y ) is separated, then (A, X) −◦(B, Y ) is separated; equivalently, the tensor product of extensional objects is extensional;
5. chu(D , R) becomes a ∗-autonomous category when we define
(A, X) −◦(B, Y ) = E((A, X) −◦(B, Y ))
(A, X) ⊗ (B, Y ) = S((A, X) ⊗ (B, Y ))
where the right hand sides of these formulas refer to the operations in Chu(D , R) and
the left hand sides define the operations in chu(D , R).
4.3. Conventions. Here and later, when S is a set and A is an object of some category,
we denote by S · A an S-fold coproduct of copies of A. When A = R, this is simply the
free R-module generated by S.
From now on, we will be restricting our attention to objects of chu(D , R), unless
explicitly stated otherwise.
4.4. Self-injective rings. In a number of cases the category of chu objects is equivalent to one or more concrete categories of topological objects. See [Barr & Kleisli (2001)]
or the unpublished note [Barr, (unpublished)] for an example that generalizes Pontrjagin
duality.
In order to motivate what follows, we begin with the special case in which R is selfinjective. This case is treated in more detail in Section 6 below. If U = (A, X) is an
Ä
object of chu(D , R), the separability condition implies that A Â / R||X|| . Let σU denote
the topological module whose underlying module is A and whose topology is such that
6
σU ⊆ R||X|| is a topological embedding, with R topologized discretely. We will see in
6.1 below that R continues to be injective in a category B that contains C as a full
subcategory; a fortiori R is injective in C with respect to topological embeddings.
We claim that σU determines U. Observe that we can view hom(σU, R) as a submodule of hom(A, R) consisting of the maps continuous in the topology induced by
R||X|| . Moreover, each element of X is continuous in that topology, so that we have
X ⊆ hom(σU, R) ⊆ hom(A, R). It follows from Corollary 3.3 that the canonical map
Ä
||X|| · R Â / hom(R||X|| , R) is an isomorphism. Injectivity of R applied to the embedding
/ hom(σU, R) is surjective. The image of this map is
/ R||X|| implies that ||X|| · R
σU
the submodule of hom(σU, R) generated by the elements of X, which is X itself, so that
X = Hom(σU, A). Thus we conclude that X ∼
= hom(σU, R) and so we recover U from
σU as (|σU|, hom(σU, R)).
This argument evidently depends on the injectivity of R. In order to deal with the
case that R is not injective (but its complete ring of quotients is) we will have to introduce
new subcategories of chu(D , R).
/ C and ρ :
The functors σ and ρ. We introduce functors σ : chu(D , R)
/ chu(D , R) as follows. If U = (A, X) is an object of chu(D , R), then σU is the
C
module A topologized as a subobject of R||X|| . If C is an object of C , let ρC = (|C|, |C ? |)
with the obvious pairing.
4.5.
4.6.
Proposition. ρ is left adjoint to σ.
/ (A, X) is given by R-linear
Proof. As seen in Diagram (∗) on Page 4, a map ρC
/ |C ? | for which the left-hand square of
/ A and X
homomorphisms |C|
|C|
/ Hom(|C ? |, R)
/ R||C ? ||
²
²
/ Hom(X, R)
²
/ R||X||
A
commutes, while the right-hand one obviously does. But the commutation of the outer
/ σ(A, X) when A is topologized by
square is the condition required for continuity of C
||X||
/ σ(A, X).
the embedding into R . Thus we get a map C
/ A for which the
/
σ(A, X) consists of a map |C|
In the other direction, a map C
/A
/ R||X|| is continuous. Dualizing gives a map X · R
/ |C ? | which
composite C
?
/ X · R gives a map X
/ |C |. This function
when composed with the inclusion X
/
/ |C ? | and the
/
X·R
Hom(A, R)
is actually a homomorphism as it factors X
composite of the first two as well as the third are homomorphisms. The remaining details
are left to the reader.
4.7.
Proposition. For any C ∈ C , the inner adjunction C
/ σρC is an isomorphism.
Proof. This follows from the facts that ρC = (|C|, |C ? |) and that σρC is just |C| topol?
ogized as a subspace of R||C || , which is just the original topology on C by 3.1.
7
Next we identify the objects of chu(D , R) on which ρσ is the identity. If U = (A, X)
is an object of chu(D , R), then σU = A, topologized as a subobject of R||X|| . Then
ρσU = (A, hom(σU, R)). Since the elements of X induce continuous maps on σU, we
/ hom(σU, R) Â Ä / hom(A, R). Since U is extensional the composite is monic
have X
and hence so is the first map. In other words, we can assume X ⊆ hom(σU, R). There
is, however, no reason to suppose that X = hom(σU, R) so that ρσ is not generally the
identity. To deal with this situation, we introduce new conditions on objects of the chu
category.
4.8. Definition. We say that U is high when ρσU = U. We also say that U is wide
when U⊥ is high.
We let chu(D , R)h , chu(D , R)w and chu(D , R)hw denote the full subcategories of
chu(D , R) consisting, respectively, of the high objects, the wide objects and the high,
wide objects.
If (A, X) is an object of chu(D , R), the topology induced on A by its embedding into
px
/ RX
/ R. It follows
RX has a subbase at 0 given by the kernels of the composites A
/
that ϕ : A
R is continuous in this topology
if and only if there are finitely many
T
elements x1 , . . . , xn ∈ X such that ker ϕ ⊇ kerh−, xi i. Thus we conclude:
4.9. Proposition. The object (A, X) ∈ chu(D , T
R) is high if and only if, for all ϕ :
/
R and all x1 , . . . , xn ∈ X such that ker ϕ ⊇ kerh−, xi i, there is an x ∈ X such
A
that ϕ = h−, xi.
4.10. Proposition. If U = (A, X) ∈ chu(D , R), then the cokernel of the map ||X|| ·
/ σU that is induced by the inclusion U ⊆ R||X|| is weak torsion.
R
Proof. This is immediate from 3.3 and 3.4.
Ä
4.11. Proposition. The inclusion chu(D , R)h  / chu(D , R) has a right adjoint H and
Ä
the inclusion chu(D , R)w  / chu(D , R) has a left adjoint W .
Proof. We claim that H = ρσ. In fact, suppose U is high and V is arbitrary. Then,
since σ is full and faithful and ρ is its left adjoint, we have
Hom(U, ρσV) ∼
= Hom(σU, σρσV) ∼
= Hom(σU, σV)
∼
= Hom(ρσU, V) ∼
= Hom(U, V)
which shows the first claim. For the second, let W V = (H(V⊥ ))⊥ .
4.12. Notation. We will denote H(A, X) by (A, X̄) since the first coordinate is A and
the second is the subset of hom(A, R) consisting of the continuous maps. That X ⊆ X̄
follows from extensionality. Similarly, we denote W (A, X) by (Ā, X). However, since, as
we will see in Example 4.19, HW is not naturally equivalent to WH, we will never use
the ambiguous (Ā, X̄) outside of this very sentence.
8
Ä
4.13. Remark. Since X Â / X̄ is essentially induced from the topological inclusion
/ RX , it follows from Theorem 3.4 that the cokernel of X Â Ä / X̄ is weak torsion.
A
Recall from 4.1 and Theorem 4.2.4 that when U = (A, X) and V = (B, Y ) are
objects of chu(D , R), the tensor product in Chu(D , R) is given by (A, X) ⊗ (B, Y ) =
(A ⊗ B, [U, V⊥ ]) and is extensional (but not generally separated). Its separated reflection
is gotten by factoring out of A ⊗ B the elements that are annihilated by every map
/ V⊥ . It nonetheless makes sense to talk of continuous maps A ⊗ B
/ R.
U
In the study of Chu objects that are separated and extensional, a crucial point was
that the separated reflection commuted with the extensional coreflection. One would
similarly hope here that the wide reflection might commute with the high coreflection.
That this fails will be shown in Example 4.19. However, the only real consequence of that
hoped-for commutation that matters to us remains true:
4.14.
Proposition. If U is high, so is W U; dually if U is wide, so is HU.
Proof. It suffices to prove the first claim. Let U = (A, X) be high, with W U = (Ā, X).
Ä
As noted above in 4.13, the cokernel of A Â / Ā is weak torsion. Since weak torsion modules
/ R that
have no non-zero homomorphisms into R, we see that two homomorphisms Ā
/ R and x1 , . . . , xn ∈ X such that
agree on TA are equal. Now suppose that
T ϕ : Ā
ker ϕ ⊇ kerh−, xi i. Then ker(ϕ|A) ⊇ ker(h−, xi i|A). Since A is wide, there is an
x ∈ X such that ϕ|A = h−, xi|A. But then ϕ = h−, xi on all of Ā.
4.15. Proposition. Let U = (A, X) and V = (B, Y ) be high and extensional. Then
U ⊗ V is high (and extensional).
Proof. By definition U ⊗ V = (A ⊗ B, [U, V⊥ ]). This means that the topology on
⊥
A ⊗ B is induced by its map to RHom(U,V ) . Note that even when U and V are separated
/ R is continuous in this topology,
this map is not necessarily injective. If ϕ : A ⊗ B
/ V⊥ such that
then, as Tin Proposition 4.9, there are maps (f1 , g1 ), . . . (fn , gn ) : U
ker ϕ ⊇ ker(fi , gi ). Here
T (fi , gi ) acts on A ⊗ B by (fi , gi )(a ⊗ b) = hb, fi ai = ha, gi bi.
Now fix a ∈ A. If b ∈ kerh−, fi ai then b ∈ ker ϕ(a, −). Since for any y ∈ Y , the map
/ R is continuous. Since B is
h−, yi is continuous on B, it follows that ϕ(a, −) : B
high, there is a unique y = f a ∈ Y such that ϕ(a ⊗ b) = hb, f ai for all b ∈ B. The usual
/ Y . There
arguments involving uniqueness show that f is an R-linear homomorphism A
/ X such that ϕ(a ⊗ b) = ha, gxi. It follows that ϕ is
is similarly an R-linear map g : B
⊥
/
in the image of [U, V ]
hom(A ⊗ B, R). Extensionality is proved in [Barr(1998)].
4.16.
Corollary. If U is high and V is wide, then U −◦ V is wide.
Proof. This is immediate from the fact that U −◦ V = (U ⊗ V⊥ )⊥ .
Now we can define the ∗-autonomous structure on chu(D , R)hw . Assume that U
and V are high and wide. Then U ⊗ V is high and U −◦ V is wide. So we define
U −−◦
V = H(U −◦ V) and U ⊗ V = W (U ⊗ V).
h
w
9
4.17.
Theorem. For any U, V, W ∈ chu(D , R)hw , we have
Hom(U ⊗ V, W) ∼
W)
= (U, V −−◦
h
w
Proof.
Hom(U ⊗ V, W) = Hom(W (U ⊗ V), W) ∼
= Hom(U ⊗ V, W)
w
∼
= Hom(U, V −◦ W) ∼
= Hom(U, H(V −◦ W))
= Hom(U, V −−◦
W)
h
Since it is evident that U −−◦
V ∼
U⊥ , we conclude that chuhw is a ∗= V⊥ −−◦
h
h
autonomous category.
4.18.
Theorem. chu(D , R)hw is complete and cocomplete.
Proof. chu(D , R)h is a coreflective subcategory of chu(D , R) and is therefore complete
and cocomplete and chu(D , R)hw is a reflective subcategory of chu(D , R)h and the same
is true of it.
4.19.
Example. In general, HW is not naturally isomorphic to WH.
Assume that R contains an element r that is neither invertible nor a zero divisor. Let
(1/r)R denote the R-submodule of the classical ring of quotients (gotten by inverting all
the non zero-divisors of R) generated by 1/r. Then as modules, we have proper inclusions
rR ⊆ R ⊆ (1/r)R.
Let U = (R, R) and V = (R, rR) both using multiplication as pairing. It is clear that
U ∈ chu(D , R)hw . As for V, it is evident that σV = R. The topology is discrete since
under the inclusion R ⊆ R||rR|| , the only element of R that goes to 0 under the projection
on the coordinate r is 0, since r is not a zero-divisor. Then HV = ρσV = (R, R).
/ V is the identity on the first coordinate and inclusion on the
Moreover if (f, g) : U
second, it is clear that H(f, g) is just the identity. Since HV is obviously also wide, we
see that WH(f, g) is also the identity.
Now let us calculate HW V. Since V⊥ = (rR, R), it is clear that σ(V⊥ ) = rR.
/ σ(U⊥ ) is the proper inclusion rR ⊆ R and is not the
Moreover σ(g, f ) : σ(V⊥ )
identity. We can identify ρσ(V⊥ ) as (rR, (1/r)R) so that W V = ((1/r)R, rR), which is
isomorphic to U and is therefore high and wide. Thus HW V = W V (assuming, as we
may, that H is the identity on chuh and W is the identity on chuw ). So WH(f, g) is an
isomorphism but HW (f, g) is not, which readily implies that WH cannot be naturally
isomorphic to HW .
5. The ∗-autonomous structure in C
Recall that for C ∈ C , C ? is the object hom(C, R) topologized as a subset of R||C|| .
10
/ C ?? that takes an element of C to evaluation at that
5.1. Proposition. The map C
element is a topological embedding.
Ä
Proof. By hypothesis, there is an embedding C Â / RS for some set S. This transposes
/ RS . The diagram
/ Hom(C, R) = ||C ? || and then R||C ? ||
to a function S
Ä
CÂ
/ RS
O
²
Ä
C ?? Â
/ R||C ? ||
commutes and the result is that the left-hand arrow, being an initial factor of an embedding, is one itself.
We say that the object C is reflexive if the canonical map C
Let Cr denote the full subcategory of reflexive objects.
5.2.
/ C ?? is an isomorphism.
Proposition. For any C ∈ C , the object C ? is reflexive.
Ä
/ C ? and composes with the canonical
Proof. We have C Â / C ?? which gives C ???
?
???
???
?
∼
/C
C
to give the identity. Thus C
= C ⊕ C 0 for a submodule C 0 ⊆ C ??? . In
/ R||C ? || gives a
particular, C 0 is weak torsion free. On the other hand, the inclusion C ??
/ C ??? whose cokernel T is, by 3.4, weak torsion. But since the canonical
map ||C ? || · R
/ C ? is obviously surjective, we conclude that T ∼
map ||C ? || · R
= C 0 , which implies that
both are zero.
5.3. Proposition. Let U ∈ chu(D , R) be high. Then σU is reflexive if and only if U
is also wide.
Proof. Let U = (A, X). In general (σU)? = (X̄, A), which is the same as (X, A) =
σ(U⊥ ) since U is assumed high. The same argument implies that (σU)?? = σ(Ā, X) and
this is σU if and only if Ā = A, that if and only if is U is also wide.
5.4. Corollary. The functors σ and ρ induce inverse equivalences between the categories chu(D , R)hw and Cr .
5.5. The internal hom in Cr . The obvious candidate for an internal hom in Cr is to
0
let C −◦ C 0 be hom(C, C 0 ) embedded as a topological subobject of R||C||×||C || . But it is not
obvious that this defines a reflexive object. With the help of the ∗-autonomous structure
on chu(D , R)hw , we can prove this.
5.6. Theorem. For C, C 0 ∈ Cr , the object C −◦ C 0 , as defined in the preceding paragraph, is reflexive. In fact, C −◦ C 0 ∼
ρC 0 ).
= σ(ρC −−◦
h
11
Proof. For this argument, when C, C 0 ∈ Cr , we let C −−◦
C 0 = σ(ρC −−◦
ρC 0 ). We wish
r
h
to show that C −−◦
C 0 = C −◦ C 0 . We have that
r
C −−◦
C 0 = σ(ρC −−◦
ρC 0 ) = σH(ρC −◦ ρC 0 )
r
h
= σρσ(ρC −◦ ρC 0 )
(because H = ρσ, 4.11)
= σ(ρC −◦ ρC 0 )
(from 4.7)
Now, we have
ρC −◦ ρC 0 = ([ρC, ρC 0 ], |C| ⊗ |C 0? |) ∼
= (hom(C, C 0 ), |C| ⊗ |C 0? |)
since ρ is full and faithful on Cr . Thus σ(ρC −◦ ρC 0 ) is just hom(C, C 0 ) equipped with
0?
the topology it inherits from R|||C|⊗|C ||| . But the definition of C −◦ C 0 is just hom(C, C 0 )
0?
equipped with the topology it inherits from R||C||×||C || so it suffices to show that those
/ |||C| ⊗ |C 0? ||| which gives rise
topologies coincide. There is a natural map ||C|| × ||C 0? ||
0?
to a map R||C⊗C || . From the commutative triangle
Ä
/ R||C||×||C 0 ||
O
OOO
OOO
OOO
OOO
‘
OOO
OOO
'
hom(C, C 0 ) Â
0
R|||C|⊗|C |||
/ R|||C|⊗|C 0 ||| , as a first factor in a topological embedding, is also
we see that hom(C, C 0 )
a topological embedding. Thus C −◦ C 0 is just σ(ρC −−◦
ρC 0 ).
h
6. The case of a self-injective ring
Things get simpler and a new possibility opens when R is a commutative self-injective
ring. This was done in [Barr (1999)] when R is a field and what was done there goes
through unchanged when R is separable (that is, a product of finitely many fields), but
there are many more rings that are self-injective, for example, an arbitrary product of
fields, and any complete boolean ring.
Throughout this section we will suppose that R is a commutative self-injective ring.
Let S and T be sets. We denote by Q(S, T ) the module RS ×|RT |. The module R itself
is always topologized discretely. We let B denote the category of topological R-modules
that can be embedded into a module of the form Q(S, T ) for some sets S and T . The
principal use we make of self-injectivity is contained in the following.
6.1.
Theorem. R is injective in B with respect to topological inclusions.
12
Proof. (Compare the proof of Theorem 3.2) It is easy to reduce this to the case of an
Ä
object embedded into an object of the form Q(S, T ). So we begin with B Â / Q(S, T ) and
/ R. The topology on Q(S, T ) has a basis at 0 of
a continuous homomorphism ϕ : B
sets of the form Q(S − S0 , T ) for a finite subset S0 ⊆ S. Then the kernel of ϕ contains
a set of the form B ∩ Q(S − S0 , T ). Let B0 = B/(B ∩ Q(S − S0 , T )). Then we get a
commutative diagram
Ä
/ Q(S, T )
BÂ
ϕ
² Ä
B0 Â
ϕ0
» ² zt t
t
t
t
t
t
²
/ Q(S0 , T )
t
t
R
in which the diagonal map exists because B0 and Q(S0 , T ) are discrete and R is injective.
6.2.
Corollary. When R is injective, every object of chu(D , R) is high and wide.
Proof. The proof is left as a simple exercise (see the discussion in 4.4).
We will say that an object of B is weakly topologized if it is embedded in a power
of R (with the product topology). Let Bwk be the full subcategory of B consisting of the
weakly topologized objects. Evidently Bwk is the same as C , as used before this section.
It is clear that every object of chu(D , R) is both high and wide and thus σ and ρ, as
defined in 4.5 are actually equivalences between Bwk and chu(D , R). Since chu(D , R) is
∗-autonomous, so is Bwk . Thus we have,
6.3. Theorem. The category Bwk of R-cogenerated topological algebras is ∗-autonomous
when B −◦ B 0 is defined as hom(B, B 0 ) topologized as subspace of B 0||B|| .
6.4. The strongly topologized objects. Let B ∈ B . We say that B is strongly
/ B is a bijection in B that induces an isomorphism
topologized if whenever B 0
0
/ Hom(B , R), then B 0
/ B is an isomorphism. We have already deHom(B, R)
noted the category of weakly topologized modules by Bwk and we denote by Bst the full
subcategory of strongly topologized modules.
For B ∈ B , let S(B) index the set of all topological R-modules that have the same
underlying R-module as B, a stronger topology and the same set of continuous homomorphisms into R.
6.5. Theorem. Let B be an object of B . Then among all objects Bs with s ∈ S(B),
there is one whose topology is finer than all the others.
13
Proof. Let τ B be defined so that
/
τB
Q
s∈S
²
/ BS
²
B
Bs
diag
Q
/ B S is a bijection, so is τ B
/ B so it suffices to show that
is a pullback. Since Bs
/
Hom(B, R)
Hom(τ B, R) is an isomorphism. It is clearly monic, so it suffices to see
that it is surjective. In this diagram, the bottom arrow is a topological embedding, hence
so is the top arrow. If we apply the functor hom(−, R) to the above diagram and use
Corollary 3.3, we get
P
hom(τ B, R) o o
O
homs∈S (Bs , R)
O
∼
=
hom(B, R) o
S · hom(B, R)
/ hom(τ B, R) is surjective.
from which we see immediately that hom(B, R)
This theorem and its proof are essentially identical to that of [Barr & Kleisli (2001),
Theorem 4.1 (proof that 2 implies 3)]. Another approach is used for the similar [Barr &
Kleisli (1999), Proposition 3.8]. The latter paper omitted, however to prove that τ was
functorial. We give the proof here, essentialy the same as that of the former citation.
6.6. Theorem. The object function τ determines a functor τ : B
adjoint to the inclusion.
/ Bst which is left
/ B is an arrow in B and Bs
Proof. It will be sufficient to show that whenever B 0
0
0
for s ∈ S(B ), then there is an s ∈ S(B ) and a commutative square
Bs0
/ B0
²
²
/B
Bs
/B
We let B 00 be a pullback Bs ×B B 0 . Since Bs has the same underlying module as B, we
can assume that B 00 has the same underlying module as B 0 . Thus it suffices to show it has
/ Bs × B 0 is an equalizer
the same set of continuous maps to R as B 0 does. The map B 00
/ B so that B 00 is embedded in the product Bs × B. Then we have
of two maps Bs × B 0
14
a commutative square
Ä
B 00 Â
/ B 0 × Bs
²
²
/ B0 × B
B0
If we apply the functor hom(−, R) we get the square
hom(B 00 , R) o o
O
hom(B 0 , R) × hom(Bs , R)
O
∼
=
hom(B 0 , R) o
hom(B 0 , R) × hom(B, R)
/ hom(B 00 , R) is surjective and it is obviously
from which it follows that hom(B 0 , R)
injective so that B 00 = Bs0 for some s0 ∈ S(B 0 ). This shows that τ is a functor and the
adjunction is clear.
If we let ωB denote the module |B| retopologized by its embedding into RHom(B,R)
then ωB is the weakest topology with the same set of homomorphisms into R as B and
τ B is the strongest. We leave it to the reader to prove the easy fact that ω is right to the
inclusion of Bwk into B . Clearly τ ω = τ and ωτ = ω.
6.7.
Theorem. τ and ω induce inverse equivalences between Bwk and Bst .
Proof. If B ∈ Bwk , then τ B ∈ Bst and ωτ B = ωB = B, while if B ∈ Bst , then ωB ∈ Bwk
and τ ωB = τ B = B.
Consequently, Bst is also ∗-autonomous. The internal hom is gotten by first forming
B −◦ B 0 and then applying τ .
7. Discussion
It is obvious that the ∗-autonomous structure on the category Cr depends crucially on 5.6.
But we could not find a proof of that fact independent of the chu category, in particular,
the high wide subcategory. While it is certainly true that you use the methods that work,
an independent argument would still be desirable.
For example, suppose R is a ring that is not necessarily commutative. The category
of two-sided R-modules has an obvious structure of a biclosed monoidal category. If
A is a topological module, it has both a left dual ∗A (consisting of the left R-linear
homomorphisms into R) and a right dual A∗ . The two duals commute and there is
/ ∗A∗ . It is natural to call an object reflexive if that map is an
a canonical map A
isomorphism. One can now ask if the internal homs (that is, the left and right homs)
of two reflexive objects is reflexive. It might require something like the left and right
15
complete rings of quotients being isomorphic and also R-injective. Even the case that R
has no zero divisors would be interesting.
Although there a Chu construction for a biclosed monoidal category, with an infinite
string of left and right duals (see [Barr, 1995]), there does not seem to be any obvious
way of defining separated or extensional objects. An object might be separated, say, with
respect to its right dual, but not its left. Factoring out elements that are annihilated by
the left dual would usually lead to the right dual being undefined. And even if a notion
of separated, extensional objects was definable, what possible functor to the topological
category would exist that would transform to all the infinite string of duals? Thus we
would require an independent argument for the analog of Theorem 5.6.
References
M. Barr (1995), Non-symmetric *-autonomous categories. Theoretical Computer Science,
139, 115—130. ftp://ftp.math.mcgill.ca/pub/barr/pdffiles/asymm.pdf
M. Barr, The separated extensional Chu category (1998). Theory Appl. Categories 4
127–137. ftp://ftp.math.mcgill.ca/pub/barr/pdffiles/bbr.pdf
M. Barr ∗-autonomous categories: Once more around the track. Theory Appl. Categories,
6 (1999), 5–24. ftp://ftp.math.mcgill.ca/pub/barr/pdffiles/omatt.pdf
M. Barr, On duality of topological abelian groups. ftp://ftp.math.mcgill.ca/pub/
barr/pdffiles/abgp.pdf
M. Barr, J. Kennison, and R. Raphael (2008), Isbell duality. Theory Appl. Categories 20,
504–542. ftp://ftp.math.mcgill.ca/pub/barr/pdffiles/isbell.pdf
M. Barr, J. Kennison, and R. Raphael (2009), Isbell duality for modules. Preprint. ftp:/
/ftp.math.mcgill.ca/pub/barr/pdffiles/rmod.pdf
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Catégorique, 40 (1999), 3-20. ftp://ftp.math.mcgill.ca/pub/barr/pdffiles/
balls.pdf
M. Barr and H. Kleisli (2001), On Mackey topologies in topological abelian groups. Theory Appl. Categories 8, 54–62. ftp://ftp.math.mcgill.ca/pub/barr/pdffiles/
mackey.pdf
J. Lambek (1986), Lectures on Rings and Modules, third edition. AMS Chelsea.
Department of Mathematics and Statistics
McGill University, Montreal, QC, H3A 2K6
Department of Mathematics and Computer Science
Clark University, Worcester, MA 01610
16
Department of Mathematics and Statistics
Concordia University, Montreal, QC, H4B 1R6
Email: [email protected], [email protected], [email protected]